Pulsatile blood flow, shear force, energy dissipation and Murray's Law

BACKGROUND: Murray's Law states that, when a parent blood vessel branches into daughter vessels, the cube of the radius of the parent vessel is equal to the sum of the cubes of the radii of daughter blood vessels. Murray derived this law by defining a cost function that is the sum of the energy cost of the blood in a vessel and the energy cost of pumping blood through the vessel. The cost is minimized when vessel radii are consistent with Murray's Law. This law has also been derived from the hypothesis that the shear force of moving blood on the inner walls of vessels is constant throughout the vascular system. However, this derivation, like Murray's earlier derivation, is based on the assumption of constant blood flow. METHODS: To determine the implications of the constant shear force hypothesis and to extend Murray's energy cost minimization to the pulsatile arterial system, a model of pulsatile flow in an elastic tube is analyzed. A new and exact solution for flow velocity, blood flow rate and shear force is derived. RESULTS: For medium and small arteries with pulsatile flow, Murray's energy minimization leads to Murray's Law. Furthermore, the hypothesis that the maximum shear force during the cycle of pulsatile flow is constant throughout the arterial system implies that Murray's Law is approximately true. The approximation is good for all but the largest vessels (aorta and its major branches) of the arterial system. CONCLUSION: A cellular mechanism that senses shear force at the inner wall of a blood vessel and triggers remodeling that increases the circumference of the wall when a shear force threshold is exceeded would result in the observed scaling of vessel radii described by Murray's Law

Comparison of four mathematical models to analyze indicator-dilution curves in the coronary circulation

While several models have proven to result in accurate estimations when measuring cardiac output using indicator dilution, the mono-exponential model has primarily been chosen for deriving coronary blood/plasma volume. In this study, we compared four models to derive coronary plasma volume using indicator dilution; the mono-exponential, power-law, gamma-variate, and local density random walk (LDRW) model. In anesthetized goats (N = 14), we determined the distribution volume of high molecular weight (2,000 kDa) dextrans. A bolus injection (1.0 ml, 0.65 mg/ml) was given intracoronary and coronary venous blood samples were taken every 0.5–1.0 s; outflow curves were analyzed using the four aforementioned models. Measurements were done at baseline and during adenosine infusion. Absolute coronary plasma volume estimates varied by ~25% between models, while the relative volume increase during adenosine infusion was similar for all models. The gamma-variate, LDRW, and mono-exponential model resulted in volumes corresponding with literature, whereas the power-model seemed to overestimate the coronary plasma volume. The gamma-variate and LDRW model appear to be suitable alternative models to the mono-exponential model to analyze coronary indicator-dilution curves, particularly since these models are minimally influenced by outliers and do not depend on data of the descending slope of the curve only

Extension of Murray's law using a non-Newtonian model of blood flow

<p>Abstract</p> <p>Background</p> <p>So far, none of the existing methods on Murray's law deal with the non-Newtonian behavior of blood flow although the non-Newtonian approach for blood flow modelling looks more accurate.</p> <p>Modeling</p> <p>In the present paper, Murray's law which is applicable to an arterial bifurcation, is generalized to a non-Newtonian blood flow model (power-law model). When the vessel size reaches the capillary limitation, blood can be modeled using a non-Newtonian constitutive equation. It is assumed two different constraints in addition to the pumping power: the volume constraint or the surface constraint (related to the internal surface of the vessel). For a seek of generality, the relationships are given for an arbitrary number of daughter vessels. It is shown that for a cost function including the volume constraint, classical Murray's law remains valid (i.e. Σ<it>R</it>^{<it>c </it>}= <it>cste </it>with <it>c </it>= 3 is verified and is independent of <it>n</it>, the dimensionless index in the viscosity equation; <it>R </it>being the radius of the vessel). On the contrary, for a cost function including the surface constraint, different values of <it>c </it>may be calculated depending on the value of <it>n</it>.</p> <p>Results</p> <p>We find that <it>c </it>varies for blood from 2.42 to 3 depending on the constraint and the fluid properties. For the Newtonian model, the surface constraint leads to <it>c </it>= 2.5. The cost function (based on the surface constraint) can be related to entropy generation, by dividing it by the temperature.</p> <p>Conclusion</p> <p>It is demonstrated that the entropy generated in all the daughter vessels is greater than the entropy generated in the parent vessel. Furthermore, it is shown that the difference of entropy generation between the parent and daughter vessels is smaller for a non-Newtonian fluid than for a Newtonian fluid.</p

Subcoronary versus supracoronary aortic stenosis. an experimental evaluation

<p>Abstract</p> <p>Background</p> <p>Valvular aortic stenosis is the most common cause of left ventricular hypertrophy due to gradually increasing pressure work. As the stenosis develop the left ventricular hypertrophy may lead to congestive heart failure, increased risk of perioperative complications and also increased risk of sudden death. A functional porcine model imitating the pathophysiological nature of valvular aortic stenosis is very much sought after in order to study the geometrical and pathophysiological changes of the left ventricle, timing of surgery and also pharmacological therapy in this patient group.</p> <p>Earlier we developed a porcine model for aortic stenosis based on supracoronary aortic banding, this model may not completely imitate the pathophysiological changes that occurs when valvular aortic stenosis is present including the coronary blood flow. It would therefore be desirable to optimize this model according to the localization of the stenosis.</p> <p>Methods</p> <p>In 20 kg pigs subcoronary (n = 8), supracoronary aortic banding (n = 8) or sham operation (n = 4) was preformed via a left lateral thoracotomy. The primary endpoint was left ventricular wall thickness; secondary endpoints were heart/body weight ratio and the systolic/diastolic blood flow ratio in the left anterior descending coronary. Statistical evaluation by oneway anova and unpaired t-test.</p> <p>Results</p> <p>Sub- and supracoronary banding induce an equal degree of left ventricular hypertrophy compared with the control group. The coronary blood flow ratio was slightly but not significantly higher in the supracoronary group (ratio = 0.45) compared with the two other groups (subcoronary ratio = 0.36, control ratio = 0.34).</p> <p>Conclusions</p> <p>A human pathophysiologically compatible porcine model for valvular aortic stenosis was developed by performing subcoronary aortic banding. Sub- and supracoronary aortic banding induce an equal degree of left ventricular hypertrophy. This model may be valid for experimental investigations of aortic valve stenosis but studies of left ventricular hypertrophy can be studied equally well by graduated constriction of the ascending aorta.</p

Quantification of fractional flow reserve based on angiographic image data

Coronary angiography provides excellent visualization of coronary arteries, but has limitations in assessing the clinical significance of a coronary stenosis. Fractional flow reserve (FFR) has been shown to be reliable in discerning stenoses responsible for inducible ischemia. The purpose of this study is to validate a technique for FFR quantification using angiographic image data. The study was carried out on 10 anesthetized, closed-chest swine using angioplasty balloon catheters to produce partial occlusion. Angiography based FFR was calculated from an angiographically measured ratio of coronary blood flow to arterial lumen volume. Pressure-based FFR was measured from a ratio of distal coronary pressure to aortic pressure. Pressure-wire measurements of FFR (FFRP) correlated linearly with angiographic volume-derived measurements of FFR (FFRV) according to the equation: FFRP = 0.41 FFRV + 0.52 (P-value < 0.001). The correlation coefficient and standard error of estimate were 0.85 and 0.07, respectively. This is the first study to provide an angiographic method to quantify FFR in swine. Angiographic FFR can potentially provide an assessment of the physiological severity of a coronary stenosis during routine diagnostic cardiac catheterization without a need to cross a stenosis with a pressure-wire

Does the principle of minimum work apply at the carotid bifurcation: a retrospective cohort study

<p>Abstract</p> <p>Background</p> <p>There is recent interest in the role of carotid bifurcation anatomy, geometry and hemodynamic factors in the pathogenesis of carotid artery atherosclerosis. Certain anatomical and geometric configurations at the carotid bifurcation have been linked to disturbed flow. It has been proposed that vascular dimensions are selected to minimize energy required to maintain blood flow, and that this occurs when an exponent of 3 relates the radii of parent and daughter arteries. We evaluate whether the dimensions of bifurcation of the extracranial carotid artery follow this principle of minimum work.</p> <p>Methods</p> <p>This study involved subjects who had computed tomographic angiography (CTA) at our institution between 2006 and 2007. Radii of the common, internal and external carotid arteries were determined. The exponent was determined for individual bifurcations using numerical methods and for the sample using nonlinear regression.</p> <p>Results</p> <p>Mean age for 45 participants was 56.9 ± 16.5 years with 26 males. Prevalence of vascular risk factors was: hypertension-48%, smoking-23%, diabetes-16.7%, hyperlipidemia-51%, ischemic heart disease-18.7%.</p> <p>The value of the exponent ranged from 1.3 to 1.6, depending on estimation methodology.</p> <p>Conclusions</p> <p>The principle of minimum work (defined by an exponent of 3) may not apply at the carotid bifurcation. Additional factors may play a role in the relationship between the radii of the parent and daughter vessels.</p

Sizing Up Allometric Scaling Theory

Metabolic rate, heart rate, lifespan, and many other physiological properties vary with body mass in systematic and interrelated ways. Present empirical data suggest that these scaling relationships take the form of power laws with exponents that are simple multiples of one quarter. A compelling explanation of this observation was put forward a decade ago by West, Brown, and Enquist (WBE). Their framework elucidates the link between metabolic rate and body mass by focusing on the dynamics and structure of resource distribution networks—the cardiovascular system in the case of mammals. Within this framework the WBE model is based on eight assumptions from which it derives the well-known observed scaling exponent of 3/4. In this paper we clarify that this result only holds in the limit of infinite network size (body mass) and that the actual exponent predicted by the model depends on the sizes of the organisms being studied. Failure to clarify and to explore the nature of this approximation has led to debates about the WBE model that were at cross purposes. We compute analytical expressions for the finite-size corrections to the 3/4 exponent, resulting in a spectrum of scaling exponents as a function of absolute network size. When accounting for these corrections over a size range spanning the eight orders of magnitude observed in mammals, the WBE model predicts a scaling exponent of 0.81, seemingly at odds with data. We then proceed to study the sensitivity of the scaling exponent with respect to variations in several assumptions that underlie the WBE model, always in the context of finite-size corrections. Here too, the trends we derive from the model seem at odds with trends detectable in empirical data. Our work illustrates the utility of the WBE framework in reasoning about allometric scaling, while at the same time suggesting that the current canonical model may need amendments to bring its predictions fully in line with available datasets

Deterministic Chaos and Fractal Complexity in the Dynamics of Cardiovascular Behavior: Perspectives on a New Frontier

Physiological systems such as the cardiovascular system are capable of five kinds of behavior: equilibrium, periodicity, quasi-periodicity, deterministic chaos and random behavior. Systems adopt one or more these behaviors depending on the function they have evolved to perform. The emerging mathematical concepts of fractal mathematics and chaos theory are extending our ability to study physiological behavior. Fractal geometry is observed in the physical structure of pathways, networks and macroscopic structures such the vasculature and the His-Purkinje network of the heart. Fractal structure is also observed in processes in time, such as heart rate variability. Chaos theory describes the underlying dynamics of the system, and chaotic behavior is also observed at many levels, from effector molecules in the cell to heart function and blood pressure. This review discusses the role of fractal structure and chaos in the cardiovascular system at the level of the heart and blood vessels, and at the cellular level. Key functional consequences of these phenomena are highlighted, and a perspective provided on the possible evolutionary origins of chaotic behavior and fractal structure. The discussion is non-mathematical with an emphasis on the key underlying concepts

Wall shear stress as measured in vivo: consequences for the design of the arterial system

Based upon theory, wall shear stress (WSS), an important determinant of endothelial function and gene expression, has been assumed to be constant along the arterial tree and the same in a particular artery across species. In vivo measurements of WSS, however, have shown that these assumptions are far from valid. In this survey we will discuss the assessment of WSS in the arterial system in vivo and present the results obtained in large arteries and arterioles. In vivo WSS can be estimated from wall shear rate, as derived from non-invasively recorded velocity profiles, and whole blood viscosity in large arteries and plasma viscosity in arterioles, avoiding theoretical assumptions. In large arteries velocity profiles can be recorded by means of a specially designed ultrasound system and in arterioles via optical techniques using fluorescent flow velocity tracers. It is shown that in humans mean WSS is substantially higher in the carotid artery (1.1–1.3 Pa) than in the brachial (0.4–0.5 Pa) and femoral (0.3–0.5 Pa) arteries. Also in animals mean WSS varies substantially along the arterial tree. Mean WSS in arterioles varies between about 1.0 and 5.0 Pa in the various studies and is dependent on the site of measurement in these vessels. Across species mean WSS in a particular artery decreases linearly with body mass, e.g., in the infra-renal aorta from 8.8 Pa in mice to 0.5 Pa in humans. The observation that mean WSS is far from constant along the arterial tree implies that Murray’s cube law on flow-diameter relations cannot be applied to the whole arterial system. Because blood flow velocity is not constant along the arterial tree either, a square law also does not hold. The exponent in the power law likely varies along the arterial system, probably from 2 in large arteries near the heart to 3 in arterioles. The in vivo findings also imply that in in vitro studies no average shear stress value can be taken to study effects on endothelial cells derived from different vascular areas or from the same artery in different species. The cells have to be studied under the shear stress conditions they are exposed to in real life

Sympatho-renal axis in chronic disease

Essential hypertension, insulin resistance, heart failure, congestion, diuretic resistance, and functional renal disease are all characterized by excessive central sympathetic drive. The contribution of the kidney’s somatic afferent nerves, as an underlying cause of elevated central sympathetic drive, and the consequences of excessive efferent sympathetic signals to the kidney itself, as well as other organs, identify the renal sympathetic nerves as a uniquely logical therapeutic target for diseases linked by excessive central sympathetic drive. Clinical studies of renal denervation in patients with resistant hypertension using an endovascular radiofrequency ablation methodology have exposed the sympathetic link between these conditions. Renal denervation could be expected to simultaneously affect blood pressure, insulin resistance, sleep disorders, congestion in heart failure, cardiorenal syndrome and diuretic resistance. The striking epidemiologic evidence for coexistence of these disorders suggests common causal pathways. Chronic activation of the sympathetic nervous system has been associated with components of the metabolic syndrome, such as blood pressure elevation, obesity, dyslipidemia, and impaired fasting glucose with hyperinsulinemia. Over 50% of patients with essential hypertension are hyperinsulinemic, regardless of whether they are untreated or in a stable program of treatment. Insulin resistance is related to sympathetic drive via a bidirectional mechanism. In this manuscript, we review the data that suggests that selective impairment of renal somatic afferent and sympathetic efferent nerves in patients with resistant hypertension both reduces markers of central sympathetic drive and favorably impacts diseases linked through central sympathetics—insulin resistance, heart failure, congestion, diuretic resistance, and cardiorenal disorders